Research On The Oscillation Of Second Order Nonlinear Differential Equations

The oscillation for solutions have studied enough since 1980s by many experts especial professor jiangguo Si,jurang Yan and quanxin Zhang. The oscillation of second order nonlinear differential equations or inequality with perturbation, with damping, with delay, with function or strongly sublinear differential equations are discussed and a seirious of theorems are established through many ten years. The results generalize and improve some known ones.In this paper, we concerned with the oscillation property and asy mptotic property for a class of nonlinear second order differential equation with perturbationthe oscillation property for a class of nonlinear second order differential inequality with dampingand oscillatory for solutions to a class of second order strongly sublinear differential equationsandby the method of analysis. The results amplified the theory of oscillation.First, in this paper, we discuss the oscillation of differential equationby the following theorems. Theorem 1 Eq x(t){(a(t)(?)(x(t))x'(t))' + p(t)x'(t) + q(t)f(x(t))}≤0 isoscillation ifWhere p(t)≥0,t≥t₀;(?)(x)≥c>0,f'(x)/(?)(x)≥α>0,x≠0.andρ[t₀,+∞)→(0,∞)ρ'(t)≥0.Theorem 2 Eq x(t){(a(t)(?)(x(t))x'(t))'+ p(t)x'(t) + q(t)f(x(t))}≤0 isoscillation ifWhere p(t)≥0,t≥t₀;(?)(x)≥c>0,f'(x)/(?)(x)≥α>0,x≠0.andρ[t₀,+∞)→(0,∞)ρ'(t)≥0.Theorem 3 IfThe non- oscillation solutions of Eq (a(t)φ(x(t))x'(t))'+Q(t,x(t))=P(t,x(t),x'(t))only have two types:Where (?)(x)f'(x)≥k>0,x≠0.Theorem 4 It is the necessary condition that Which made the solution x(t) of (a(t)(?)(x(t))x'(t))'+Q(t)x'(t))=P(t,x(t),x'(t))is non-oscillation type A_c.Where 0<(?)(x)<α,(?)(x)f'(x)≥k>0,x≠0 and T is large sufficient (T≥t₀).Theorem 5 It is the necessary condition thatWhich made the solution (a(t)φ(x(t))x'(t))'+Q(t,x(t))=P(t,x(t),x'(t))is non-oscillation type A₀.where 0<(?)(x)<α,(?)(x)f'(x)≥k>0,x≠0 and T is large sufficient (T≥t₀).and(?)ε<0,Theorem 1,2 generalize and reform the known results. Especially ,whenρ(t)=1,P(t,x(t),x'(t))=0,Q(t,x(t))=g(t)f(x(t)),theorem 1 is the theorem 2 inarticle [1], theorem 2 is the theorem 3 in article [1].Second, this paper offers two oscillations property of a class of differential inequalityTheorem 6 If (?)(x)f'(x)≥k>0,x≠0,and There isρ(t):[t₀,+∞)→(0,+∞), which made (a(t)ρ(t))'≥0,(?) sup a(t)ρ(t)/(?)ρ(s)ds<+∞and (?)1/(?)ρ(s)ds (?)ρ(s)(?)(q(Ï„)-p²(Ï„)/4ka(Ï„)dÏ„ds=+∞.x(t){(a(t)(?)(x(t))x'(t))'+p(t)x'(t)+g(t)f(x(t))}≤0 is oscillatory.Theorem 7 If (?)(x)f'(x)≥k>0,x≠0,0<(?) (?)(u)/f(u) du <+∞,(?) (?)(u)/f(u) du>-∞and (?) M/a(s) ds-(?) 1/a(s) (?)[q(Ï„)-p²(Ï„)/4ka(Ï„)dÏ„ds=-∞((?)M∈R, M is constant).Theorem 6 and Theorem 7 generalize and reform the known results. in article [23] and [24].Last, in this paper, the oscillation for second order strongly sublinear differential equationsandare studied according to the following thorems.Theorem 8 Ifandwhere x, y > 0, k is positive constant; and x"(t)+p(t)k(t,x(t),x'(t))x'(t)+q(t)f(x(σ(t)))g(x'(t))=0 is oscillatory.Theorem 9 Ifandx"(t)+p(t)k(t,x(t),x'(t))x'(t)+q(t)f(x(t))g(x'(t))=0 is oscillatory.Theorem 8 generalized the theorem 1 in article [2].